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Dec 16, 2023 at 23:20 comment added Igor Makhlin @Yromed To be fair, it is no coincidence that these polytopes admit nice subdivisions. A pretty popular theme are the so-called mixed subdivisions of Minkowski sums. For sums of simplices these are just subdivisions with each cell a sum of faces of the summands. I don't know much about this but $P$ should admit many such subdivisions.
Dec 16, 2023 at 17:48 vote accept Yromed
Dec 16, 2023 at 17:47 comment added Yromed Thank you very much for taking the time to write such a detailed answer. I was trying to work with unions with the hope of obtaining families of toric hypersurfaces degenerating to a union of toric divisors, but I see now why I rather consider Minkowski sums.
Dec 15, 2023 at 16:52 history edited Igor Makhlin CC BY-SA 4.0
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Dec 15, 2023 at 16:45 history edited Igor Makhlin CC BY-SA 4.0
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Dec 15, 2023 at 16:40 history edited Igor Makhlin CC BY-SA 4.0
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Dec 15, 2023 at 16:29 history answered Igor Makhlin CC BY-SA 4.0